Optimal Auctions for Correlated Buyers with Sampling Nima Haghpanah - PowerPoint PPT Presentation

Optimal Auctions for Correlated Buyers with Sampling Nima Haghpanah (Penn State) Joint with Hu Fu (UBC), Jason Hartline (Northwestern), Robert Kleinberg (Cornell) October 18, 2017 1 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent This work: ◮ Seller does not know distribution: Observes samples (past auctions) 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent This work: ◮ Seller does not know distribution: Observes samples (past auctions) ◮ Learning + CM approach fails 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent This work: ◮ Seller does not know distribution: Observes samples (past auctions) ◮ Learning + CM approach fails ◮ Full surplus extraction with “enough” information 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent This work: ◮ Seller does not know distribution: Observes samples (past auctions) ◮ Learning + CM approach fails ◮ Full surplus extraction with “enough” information 1. Extension of CM approach 2. Characterize number of samples required 2 / 21

Overview Auctions: ◮ Cremer-McLean ’88: Full surplus extraction “generically” possible ◮ Full knowledge of distribution: auction heavily detail-dependent This work: ◮ Seller does not know distribution: Observes samples (past auctions) ◮ Learning + CM approach fails ◮ Full surplus extraction with “enough” information 1. Extension of CM approach 2. Characterize number of samples required ◮ Idea: ◮ Samples as randomization device, not learning 2 / 21

The Model 1 item, n buyers (quasilinear, finite type space) 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s u i ( v i → v i , v − i , s ) ≥ u i ( v i → v ′ ∀ v i , v ′ ◮ DSIC: i , v − i , s ), i , v − i , s 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s u i ( v i → v i , v − i , s ) ≥ u i ( v i → v ′ ∀ v i , v ′ ◮ DSIC: i , v − i , s ), i , v − i , s ◮ Interim IR: E v ∼ f j , s ∼ g j [ u i ( v i → v i , v − i , s ) | v i ] ≥ 0, ∀ v i , j 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s u i ( v i → v i , v − i , s ) ≥ u i ( v i → v ′ ∀ v i , v ′ ◮ DSIC: i , v − i , s ), i , v − i , s ◮ Interim IR: E v ∼ f j , s ∼ g j [ u i ( v i → v i , v − i , s ) | v i ] ≥ 0, ∀ v i , j ◮ Goal: E v ∼ f j , s ∼ g j [revenue] = E v ∼ f j [maximum value], ∀ j 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s u i ( v i → v i , v − i , s ) ≥ u i ( v i → v ′ ∀ v i , v ′ ◮ DSIC: i , v − i , s ), i , v − i , s ◮ Interim IR: E v ∼ f j , s ∼ g j [ u i ( v i → v i , v − i , s ) | v i ] ≥ 0, ∀ v i , j ◮ Goal: E v ∼ f j , s ∼ g j [revenue] = E v ∼ f j [maximum value], ∀ j Special Case, Cremer-McLean: F = { f } (signal redundant) 3 / 21

The Model 1 item, n buyers (quasilinear, finite type space) Joint distribution f : ( v 1 , . . . , v n ) = v ∼ f . ◮ Buyers and seller know that F = { f 1 , . . . , f m } contains f ◮ f = f j for unknown j ∈ { 1 , . . . , m } ◮ Signal s drawn from g j , independent from v ◮ Example: s = ( s 1 , . . . , s k ) i.i.d samples from f j Mechanism specifies (allocation, payment) given v , s u i ( v i → v i , v − i , s ) ≥ u i ( v i → v ′ ∀ v i , v ′ ◮ DSIC: i , v − i , s ), i , v − i , s ◮ Interim IR: E v ∼ f j , s ∼ g j [ u i ( v i → v i , v − i , s ) | v i ] ≥ 0, ∀ v i , j ◮ Goal: E v ∼ f j , s ∼ g j [revenue] = E v ∼ f j [maximum value], ∀ j Special Case, Cremer-McLean: F = { f } (signal redundant) From definitions: if FSE possible for F , also possible for any { f j } 3 / 21

{ 0 , 1 } Examples 0 1 0 1 � � � � f 1 = f 2 = 0 1 / 4 1 / 8 0 1 / 4 3 / 8 1 3 / 8 1 / 4 1 1 / 8 1 / 4 4 / 21

{ 0 , 1 } Examples Cremer & McLean : ◮ If F = { f 1 } , ∃ mechanism extracts full surplus ◮ If F = { f 2 } , ∃ mechanism extracts full surplus 0 1 0 1 � � � � f 1 = f 2 = 0 1 / 4 1 / 8 0 1 / 4 3 / 8 1 3 / 8 1 / 4 1 1 / 8 1 / 4 4 / 21

{ 0 , 1 } Examples Cremer & McLean : ◮ If F = { f 1 } , ∃ mechanism extracts full surplus ◮ If F = { f 2 } , ∃ mechanism extracts full surplus 0 1 0 1 � � � � f 1 = f 2 = 0 1 / 4 1 / 8 0 1 / 4 3 / 8 1 3 / 8 1 / 4 1 1 / 8 1 / 4 4 / 21

{ 0 , 1 } Examples Cremer & McLean : ◮ If F = { f 1 } , ∃ mechanism extracts full surplus ◮ If F = { f 2 } , ∃ mechanism extracts full surplus If F = { f 1 , f 2 } , signal: i.i.d samples from the true distribution 0 1 0 1 � � � � f 1 = f 2 = 0 1 / 4 1 / 8 0 1 / 4 3 / 8 1 3 / 8 1 / 4 1 1 / 8 1 / 4 4 / 21

{ 0 , 1 } Examples Cremer & McLean : ◮ If F = { f 1 } , ∃ mechanism extracts full surplus ◮ If F = { f 2 } , ∃ mechanism extracts full surplus If F = { f 1 , f 2 } , signal: i.i.d samples from the true distribution ◮ Will see : if #samples ≥ 1, ∃ mechanism extracts full surplus 0 1 0 1 � � � � f 1 = f 2 = 0 1 / 4 1 / 8 0 1 / 4 3 / 8 1 3 / 8 1 / 4 1 1 / 8 1 / 4 4 / 21

Main Result ◮ F = { f 1 , . . . , f m } 5 / 21

Main Result ◮ F = { f 1 , . . . , f m } ◮ Recall: if FSE possible for F , then FSE possible for all { f j } 5 / 21

Main Result ◮ F = { f 1 , . . . , f m } ◮ Recall: if FSE possible for F , then FSE possible for all { f j } ◮ f a CM distribution if FSE possible for { f } 5 / 21

Main Result ◮ F = { f 1 , . . . , f m } ◮ Recall: if FSE possible for F , then FSE possible for all { f j } ◮ f a CM distribution if FSE possible for { f } ◮ dimension of F : dimension of linear space spanned by � f 1 , . . . , � f m 5 / 21